Recently, the preimages of the \( p- \) linearized polynomial \( \sum _{i=0}^{t}\alpha _{i}X^{p^i} \) over \( {\mathbb {F}}_{p} \) have been explicitly represented over \( {\mathbb {F}}_{p^n} \) for any prime p and any integer n. This paper presents an explicit representation for preimages of \( X^{q^2}+aX^q+bX \) over \( {\mathbb {F}}_{2^n} \) , i.e., for the solutions of \( X^{q^2}+aX^q+bX=c \) over \( {\mathbb {F}}_{2^n} \) for any integer n, where \( q=2^k \) for any integer k. This result pushes even further the study of the expression of solutions to affine equations over finite fields in terms of the coefficients of these equations.
